By the message I've received from you, Volk, I believe that I'm allowed (or even encouraged) to continue my little mathematical-historical lecture here. You are not aware what kind of demon you've awaken by encouraging me to speak there
Are you aware, that your example with Battle of Austerlitz is totally wrong? It is wrong not because you've based your equation on this battle - it's wrong, for you use your own equation to calculate rolls of both sides, and then you claim that they don't fit into my equation. You start with your equation, then calculate roll out of it, and then once more put this roll into the same equation. No matter what battle you take, it will seem correct, for you calulate totally nothing. Your whole calculations can be shortened to 1=1, so they are true no matter what strengths and loses you put at the beginning, no matter what you do. You basically use your own equation to prove that it's correct. So whole your proof is based on the assumption, that it's correct. You assume that your equation is correct, and then you use this assumption to prove that it's correct. That's something that cannot be done in math, it's middle school level of mathematics. Below I present you mathematical proof, that your calculations are wrong.
Spoiler
I'm using here following symbols:
A - strenght of one army before the battle
B- strenght of second army before the battle
x_A - roll of army A
x_B - roll of army B
A' - strenght of army A after the battle
B' - strenght of army B after the battle
A' = A - (A * x_b) that's your basic equation to calculate number of troops after the battle
A - A' = A * x_b just some transfomations
x_b = A/A - A'/A that's the very equation that gave you rolls of 42 and 13 for the battle of Austerlitz
now let's place it back in your original equation (so we'll do what you've done to "calculate" loses of both sides and "prove" your equation to be correct)
A' = A - (A * x_b) we start with basic equation of course and put there calculated x_b
A' = A - (A * (A/A - A'/A))
A' = A - (A - A') we've multiplied the inner bracket
A' = A - A + A' we're almost there
A' = A' no matter what numbers we put in your logic, result is always A' = A', so it's correct no matter of used numbers. Thus you've proved nothing.
But I've done something else, about what I've not thought before. I've indeed simulated 100 000 battles using our both equations, input data from battle of Austerlitz and pseudo-random generator to simulate D30 rolls. Then I've calculated avarage result of the battle using both equations. Below I present the results of this simulation
Spoiler
Firstly I've simulated it using D30 only, without any additional bonuses
France by Volk: 56293
Russia by Volk: 72495
France by Raddeo: 53637
Russia by Raddeo: 74311
France by History: 58000
Russia by History: 49400
How we can see, none of our equations can simulate results of this battle.
Secondly, I've added proposed bonuses - with addition for your equation, and multiplying for mine. I've assumed that it's summer, leaders are Bonaparte and Alexander, battle takes place outside Russia and only France uses new tactics. So to sum up, +12 for France and - 9 for Russia in your equation, and *1.4 for France and *0.7 for Russia in mine. Here are the results.
France by Volk: 67041
Russia by Volk: 60892
France by Raddeo: 57713
Russia by Raddeo: 70417
France by History: 58000
Russia by History: 49400
My equation achieved to quite well calculate loses of French side, but failed to calulate loses of bigger Russian army. Your equation gave almost no loses to France (due to huge negative "bonus" for Russia), but achieved to simulate overall proportion of troops after the battle.
But to gain better view at the real ability of equations to simulate battle I've performer several additional tests, this time only in the second version, using all bonuses.
Battle of Leipzig
France by Volk: 199603
Coalition by Volk: 271942
France by Raddeo: 172817
Coalition by Raddeo: 327569
France by History: 167000
Coalition by History: 326000
My equation almost perfectly calculated forces after the battle. Yours missed a bit more and gave better results to the weaker side. I gave bonuses for Napoleon and Tactics to France, and reduced roll for Coalition for Russians outside Russia and for Alexander, but gave them +1 (according to old notation) for Schwazenberg and austrian units (to make them a little stronger than clear russian army).
Battle of Jena-Auerstedt
France by Volk: 57253
Prussia by Volk: 89445
France by Raddeo: 48599
Prussia by Raddeo: 105793
France by History: 60170
Prussia by History: 82500
This time your equation achieved to better simulate the battle, results after battle are quite close to real outcome. My equation created bigger gap between fighting armies.
Battle of Landshut
Austria by Volk: 30989
France by Volk: 55351
Austria by Raddeo: 21253
France by Raddeo: 69046
Austria by History: 26500
France by History: 74000
My equation was much closer to the historical results, as it was battle won by stronger force which is always prefered by my equation.
Finally, battle of Waterloo
France by Volk: 54259
UK by Volk: 86498
France by Raddeo: 49642
UK by Raddeo: 102676
France by History: 32000
UK by History: 94000
This time, both our results have missed a bit. But mine gave bigger advantage to winning side, while your helped losing one.
So as we can clearly see - it's not possible to correctly calculate results of every battle (or even just major battles) with such simple equations. Results of battle depends on multiple factors - used tactics, generals, technology, weather, terrain, condition of troops, and many, many more. If you wish Volk, I could try to mathematically create such equation basing it on multiple historical battles, but job of GM to balance the factions then would be indeed terrible. For now, all those factors are simulated by D30 roll. Differently, amount of troops is not random, so it should be in logical way included into the mechanics. So as we cannot simulate result of battle with our equations, we shouldn't try to argue which equation can do it better. So should argue about which is more logical. And it is not logical to punish player for having bigger army (as your equation do). Historically bigger army was
always advantage. But other factors may also change the result of battle. Russians won battles against Persians because they had better technology, better tactics, better trained troops,
not because they used fewer troops. Using more troops would allow them to win these battles even easier. And according to your equation having bigger army is disadvantage what is irrational. If you want to realistically simulate the battles you should work a bit more on the bonuses for technology, tactics and generals. But that may easily destroy the balance of the game.
